All questions of Quadrilaterals for Class 9 Exam

The figure formed by joining the mid-points of consecutive sides of a quadrilateral is a Parallelogram.

Explanation:
A quadrilateral is a polygon with four sides. Let's consider a quadrilateral ABCD, where AB, BC, CD, and DA are the four sides.

Now, let's join the mid-points of the consecutive sides of the quadrilateral. Let the midpoints of AB, BC, CD, and DA be E, F, G, and H respectively.

To prove that the figure formed is a parallelogram, we need to show that opposite sides are parallel and equal in length.

1. Opposite sides are parallel:
- Join EF and GH. These diagonals divide the quadrilateral ABCD into four triangles: AEF, BFG, CGH, and DHG.
- By the Midpoint Theorem, EF is parallel to AB and GH is parallel to CD.
- Similarly, EG is parallel to AD and FH is parallel to BC.
- Therefore, opposite sides EF and GH are parallel, and opposite sides EG and FH are parallel.

2. Opposite sides are equal in length:
- By the Midpoint Theorem, EF = 1/2 AB and GH = 1/2 CD.
- Similarly, EG = 1/2 AD and FH = 1/2 BC.
- Therefore, opposite sides EF and GH are equal in length, and opposite sides EG and FH are equal in length.

Since the figure has opposite sides parallel and equal in length, it satisfies the definition of a parallelogram.

Hence, the figure formed by joining the mid-points of consecutive sides of a quadrilateral is a parallelogram.

Note: It is important to note that the converse of this statement is also true. That is, if a quadrilateral is a parallelogram, then the midpoints of its sides will form a parallelogram.

B) 60

There are several ways to approach this problem, but one of the most straightforward methods is to draw a square and label its sides and midpoints. Let's go through the solution step by step.

Step 1: Draw a square
Start by drawing a square with all sides of equal length. Label the four corners as A, B, C, and D.

Step 2: Label the midpoints
Next, label the midpoints of each side of the square. Let's call the midpoint on AB as E, BC as F, CD as G, and DA as H.

Step 3: Join the midpoints
Now, join the midpoints consecutively. That is, join E and F, F and G, G and H, and finally, H and E.

Step 4: Observe the resulting figure
Take a moment to observe the resulting figure formed by joining the consecutive midpoints. You will notice that it is a smaller square inside the original square.

Step 5: Identify the shape
Based on our observation, we can conclude that the resulting figure obtained from joining the consecutive midpoints of the sides of a square is another square. Therefore, the correct answer is option 'B' - Square.

Explanation:
When we join the consecutive midpoints of the sides of a square, we are essentially connecting the midpoints of each side. This creates a smaller square inside the original square. This smaller square shares the same center and orientation as the original square, but its sides are shorter in length. Therefore, the resulting figure is another square. This can be proven mathematically as well using properties of similar triangles and the fact that the diagonals of a square bisect each other at right angles.

In conclusion, the correct answer is option 'B' - Square.

Explanation:
To understand why the figure formed by joining the mid-points of the adjacent sides of a square is a square, let's consider a square ABCD.

Definition:
- A square is a quadrilateral with all four sides equal in length and all four angles equal to 90 degrees.

Construction:
- Let P, Q, R, and S be the midpoints of AB, BC, CD, and DA respectively.
- Join PQ, QR, RS, and SP.

Proof:
- To prove that the figure formed by joining the mid-points of the adjacent sides of a square is a square, we need to show that all four sides are equal in length and all four angles are equal to 90 degrees.

All four sides are equal in length:
- In a square ABCD, all four sides are equal. Therefore, AB = BC = CD = DA.
- In the figure formed by joining the mid-points of the adjacent sides of the square, we have PQ || AB, QR || BC, RS || CD, and SP || DA.
- By the midpoint theorem, we know that PQ = QR = RS = SP.
- Therefore, all four sides of the figure formed by joining the mid-points are equal in length.

All four angles are equal to 90 degrees:
- In a square ABCD, all four angles are equal to 90 degrees.
- In the figure formed by joining the mid-points of the adjacent sides of the square, we can see that PQ is parallel to AB and QR is parallel to BC.
- Since AB and BC are perpendicular to each other, PQ and QR are also perpendicular to each other.
- Similarly, RS is perpendicular to CD and SP is perpendicular to DA.
- Therefore, all four angles of the figure formed by joining the mid-points are equal to 90 degrees.

Conclusion:
- Since all four sides of the figure formed by joining the mid-points of the adjacent sides of a square are equal in length and all four angles are equal to 90 degrees, the figure is a square.
- Hence, option 'D', square, is the correct answer.

Given information:
- Perimeter of the parallelogram = 24 cm
- Length of the longer side = 8 cm

To find:
- Length of the shorter side

Solution:
1. A parallelogram has opposite sides of equal length. Therefore, if the longer side measures 8 cm, the opposite side will also measure 8 cm.
2. Let's assume the length of the shorter side is 'x' cm.
3. The perimeter of a parallelogram is the sum of all its sides. In this case, the perimeter is given as 24 cm.
4. The formula for the perimeter of a parallelogram is: Perimeter = 2 * (Length of longer side + Length of shorter side).
5. Substituting the given values, we have: 24 = 2 * (8 + x).
6. Simplifying the equation, we get: 24 = 16 + 2x.
7. Subtracting 16 from both sides of the equation, we get: 8 = 2x.
8. Dividing both sides of the equation by 2, we get: x = 4.
9. Therefore, the length of the shorter side is 4 cm.

Conclusion:
The measure of the shorter side of the parallelogram is 4 cm, which corresponds to option A.